The maximum value of the determinant of the matrix \(\left[\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4…
The maximum value of the determinant of the matrix
\(\left[\begin{array}{ccc}
1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \\
\sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \\
\sin ^2 x & \cos ^2 x & 1+4 \sin 2 x
\end{array}\right] \text { is }\)
0
2
4
6
Solution
Given,
\(\left|\begin{array}{ccc}
1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \\
\sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \\
\sin ^2 x & \cos ^2 x & 1+4 \sin 2 x
\end{array}\right|\)
Applying \(R_1 \rightarrow R_1-R_3\) and \(R_2 \rightarrow R_2-R_3\), we get
\(=\left|\begin{array}{ccc}
1 & 0 & -1 \\
0 & 1 & -1 \\
\sin ^2 x & \cos ^2 x & 1+4 \sin 2 x
\end{array}\right|\)
Applying \(C_2 \rightarrow C_2+C_1\)
\(=\left|\begin{array}{ccc}
1 & 1 & -1 \\
0 & 1 & -1 \\
\sin ^2 x & 1 & 1+4 \sin 2 x
\end{array}\right|\)
Applying \(R_2 \rightarrow R_2-R_1\) and \(R_3 \rightarrow R_3-R_1\)
\(=\left|\begin{array}{ccc}
1 & 1 & -1 \\
-1 & 0 & 0 \\
-1+\sin ^2 x & 0 & 2+4 \sin 2 x
\end{array}\right|=2+4 \sin 2 x\)
Since, maximum value of \(\sin 2 x\) is 1.
\(=2+4=6\)